= Solution
On the product of two sigma-finite measure spaces, <Tonelli's theorem> states that a nonnegative <measurable function> has measurable iterated-integral functions and
$$
\int_{X\times Y}f\,d(\mu\times\nu)=\int_X\int_Yf\,d\nu\,d\mu=\int_Y\int_Xf\,d\mu\,d\nu,
$$
with values allowed to be infinite. Fubini's theorem states that if $f$ is measurable and $\int_{X\times Y}|f|<\infty$, then its sections are <integrable> <almost everywhere>, both iterated integrals exist as finite integrals, and the same equality holds, also for real or complex signed functions. Tonelli applied to $|f|$ can first be used to verify the absolute-integrability hypothesis of Fubini.
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